Бір теңдеуді шешу үшін сандық әдіс: Қате жағдай әдісі

Numerical method used to approximate solutions of univariate equations

In mathematics, the regula falsi, method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and error technique of using test ("false") values for the variable and then adjusting the test value according to the outcome. This is sometimes also referred to as "guess and check". Versions of the method predate the advent of algebra and the use of equations. As an example, consider problem 26 in the Rhind papyrus, which asks for a solution of (written in modern notation) the equation 1=x + x/4 = 15. This is solved by false position. First, guess that 1=x = 4 to obtain, on the left, 1=4 + 4/4 = 5. This guess is a good choice since it produces an integer value. However, 4 is not the solution of the original equation, as it gives a value which is three times too small. To compensate, multiply x (currently set to 4) by 3 and substitute again to get 1=12 + 12/4 = 15, verifying that the solution is 1=x = 12. Modern versions of the technique employ systematic ways of choosing new test values and are concerned with the questions of whether or not an approximation to a solution can be obtained, and if it can, how fast can the approximation be found.

Though regula falsi always converges, usually considerably faster than bisection, there are situations that can slow its convergence – sometimes to a prohibitive degree. That problem isn't unique to regula falsi: Other than bisection, all of the numerical equation solving methods can have a slow convergence or no convergence problem under some conditions. Sometimes, Newton's method and the secant method diverge instead of converging – and often do so under the same conditions that slow regula falsi's convergence. But, though regula falsi is one of the best methods, and even in its original un improved version would often be the best choice; for example, when Newton's isn't used because the derivative is prohibitively time consuming to evaluate, or when Newton's and Successive Substitutions have failed to converge. Regula falsi's failure mode is easy to detect: The same end point is retained twice in a row. The problem is easily remedied by picking instead a modified false position, chosen to avoid slowdowns due to those relatively unusual unfavorable situations. A number of such improvements to regula falsi have been proposed; two of them, the Illinois algorithm and the Anderson–Björk algorithm, are described below.

When solving one equation, or just a few, using a computer, the bisection method is an adequate choice. Although bisection isn't as fast as the other methods—when they're at their best and don't have a problem—bisection nevertheless is guaranteed to converge at a useful rate, roughly halving the error with each iteration – gaining roughly a decimal place of accuracy with every 3 iterations. For manual calculation, by calculator, one tends to want to use faster methods, and they usually, but not always, converge faster than bisection. But a computer, even using bisection, will solve an equation, to the desired accuracy, so rapidly that there's no need to try to save time by using a less reliable method—and every method is less reliable than bisection. An exception would be if the computer program had to solve equations very many times during its run. Then the time saved by the faster methods could be significant. Then, a program could start with Newton's method, and, if Newton's isn't converging, switch to regula falsi, maybe in one of its improved versions, such as the Illinois or Anderson–Björck versions. Or, if even that isn't converging as well as bisection would, switch to bisection, which always converges at a useful, if not spectacular, rate. When the change in y has become very small, and x is also changing very little, then Newton's method most likely will not run into trouble, and will converge. So, under those favorable conditions, one could switch to Newton's method if one wanted the error to be very small and wanted very fast convergence.